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A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).

%I A376655 #14 Oct 08 2024 13:36:41
%S A376655 1,2,3,5,6,30,61,150,514,1025,5153,13390,13391,131964,502651,664312,
%T A376655 4387185,5392318,20613826
%N A376655 Sorted positions of first appearances in the second differences of consecutive squarefree numbers (A005117).
%C A376655 Warning: Do not confuse with A246655 (prime-powers exclusive).
%e A376655 The squarefree numbers (A005117) are:
%e A376655   1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, ...
%e A376655 with first differences (A076259):
%e A376655   1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, ...
%e A376655 with first differences (A376590):
%e A376655   0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, ...
%e A376655 with sorted first appearances at (A376655):
%e A376655   1, 2, 3, 5, 6, 30, 61, 150, 514, 1025, 5153, 13390, 13391, ...
%t A376655 q=Differences[Select[Range[1000],SquareFreeQ],2];
%t A376655 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376655 For first differences we had A376311 (first appearances in A076259).
%Y A376655 These are the sorted positions of first appearances in A376590.
%Y A376655 For prime-powers instead of squarefree numbers we have A376653/A376654.
%Y A376655 For primes instead of squarefree numbers we have A376656.
%Y A376655 A000040 lists the prime numbers, differences A001223.
%Y A376655 A005117 lists squarefree numbers, complement A013929 (differences A078147).
%Y A376655 A073576 counts integer partitions into squarefree numbers, factorizations A050320.
%Y A376655 For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
%Y A376655 For squarefree: A376591 (inflections and undulations), A376592 (nonzero curvature).
%Y A376655 Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.
%K A376655 nonn,more
%O A376655 1,2
%A A376655 _Gus Wiseman_, Oct 07 2024
%E A376655 a(14)-a(19) from _Chai Wah Wu_, Oct 07 2024